In
vector calculus, the
Laplace operator or
Laplacian is a
differential operator[?]. It is equal to the sum of all the second
partial derivatives of a dependent variable.
This corresponds to div(grad φ), hence the use of the symbol del to represent it:
- <math>\nabla^2 \phi = \nabla \cdot ( \nabla \phi )</math>
It is also written as Δ.
In two dimensional Cartesian coordinates, the Laplacian is:
- <math>\nabla^2 = {\partial^2 \over \partial x^2 } +
{\partial^2 \over \partial y^2 } </math>
In three:
- <math>\nabla^2 =
{\partial^2 \over \partial x^2 } +
{\partial^2 \over \partial y^2 } +
{\partial^2 \over \partial z^2 }
</math>
It occurs, for example, in Laplace's equation.